Localized patterns in star networks of Class B lasers with optoelectronic feedback
Joniald Shena, Johanne Hizanidis, Nikos E. Kouvaris, Giorgos P. Tsironis
LLocalized patterns in star networks of Class B lasers withoptoelectronic feedback
J. Shena,
1, 2
J. Hizanidis,
1, 2, ∗ N. E. Kouvaris, and G. P. Tsironis
1, 21
Department of Physics, University of Crete, 71003 Heraklion, Greece National University of Science and Technology MISiS,Leninsky prosp. 4, Moscow, 119049, Russia Department of Mathematics, Namur Institute for Complex Systems (naXys),University of Namur, Rempart de la Vierge 8, B 5000 Namur, Belgium (Dated: July 12, 2018)
Abstract
We analyze how a star network topology shapes the dynamics of coupled CO lasers with anintracavity electro-optic modulator that exhibit bistability. Such a network supports spreadingand stationary activation patterns. In particular, we observe an activation spreading where theactivated periphery turns on the center element, an activated center which drifts the periphery intothe active region and an activation of the whole system from the passive into the active region.Pinned activation, namely activation localized only in the center or the peripheral elements is alsofound. Similar dynamical behavior has been observed recently in complex networks of coupledbistable chemical reactions. The current work aims at revealing those phenomena in laser arrays,giving emphasis on the essential role of the coupling structure in fashioning the overall dynamics. ∗ [email protected] a r X i v : . [ n li n . C D ] J u l . INTRODUCTION Solid-state and semiconductor laser arrays constitute a wide family of nonlinear cou-pled systems with complex dynamic behavior. Although the emission from the individualunits is often unstable with large amplitude chaotic pulsations [1–3], the coupled systemcan show synchronization and other spatiotemporal phenomena [4]. The main differencebetween semiconductor and solid-state lasing media lies in the value of the linewidth en-hancement factor a , which is 3 ≤ a ≤ a = 0 for solid-state systems.This difference makes solid-state lasers more suitable in applications where phase locking isrequired.In recent years, there have been many studies concerning semiconductor lasers and theanalysis of synchronization and chimera states [5–8]. Here though, we focus on solid-statelaser arrays and the formation of localized stationary patterns of activity. The dynamicbehavior of each laser element is bistable and the coupling between the elements is local andarises due to the overlap of the electric fields of each separated beam [2, 9]. The theoreticalmodel we use is originated from numerical and experimental studies of a CO laser withan intracavity electro-optic modulator that exhibits bistability [10]. This model has manysimilarities to that obtained by semiconductor lasers with a saturable absorber inside thecavity [11]. A similar problem was revisited for a Nd:YAG laser with an acousto-opticmodulator [12]. Bistability were found in semiconductor lasers with strong optical injection[13] and in semiconductor laser diodes with a saturable absorber [14].Other classical examples where bistable behavior is encountered are dynamical processesin chemical systems [15, 16]. Recently, studies on complex networks of coupled bistablechemical reactions revealed rich collective dynamics, such as spreading or retreating of aninitial activation, but more interestingly, the formation of localized stationary patterns de-pendent on the coupling strength and the degree distribution of the nodes [17–20]. Beyondthe simplified theoretical approach, electrochemical experiments [19, 20] have stressed thatthe coupling topology plays a significant role in the observed dynamics resulting in a robustpattern formation mechanism. Therefore, similar findings are expected to be seen in laserarrays coupled in such a way thus forming complex networks.Here we focus on the simple case of star networks where each bistable element is connectedto a central one, the hub. This connectivity structure is often found in many natural or2ngineered systems that consist of dynamical elements interacting with each other througha common medium. It has also been used in optically coupled semiconductor lasers [21, 22]where synchronization phenomena were investigated. We present an extended numericalanalysis that takes advantage of the simplicity of the star network topology to determinethe conditions required for the formation of localized stationary patterns. We start ourinvestigation by analyzing the dynamics and determining the bistable regime for a singlelaser. In the bistable regime the active and the passive states of the laser coexist. Knowingthis we explore the dynamics of two coupled bistable lasers, before we proceed to our mainstudy for a star network of such elements. This work comments on the formation mechanismof stationary patterns which –like in the electrochemical networks– is strongly dependent onthe role of the coupling topology. II. THE MODEL
The dynamical behavior of the CO laser with feedback can be described by three coupledfirst-order differential equations, one for the laser field ( E ), the second for the populationinversion ( G ) and the last for the feedback voltage of the electro-optic modulator ( V ). Indimensionless form, the evolution equations have the form [10] dEdt = 12 (cid:0) G − − a sin ( V ) (cid:1) E (1a) dGdt = γ ( P − G − G | E | ) (1b) dVdt = β ( B + f | E | − V ) , (1c)where | E | is the amplitude of electric field, γ denotes the population decay time, P denotesthe pumping and a scales the maximum loss introduced by the modulator. The damping rate β of the feedback loop is normalized to the cavity decay rate, B is the bias voltage appliedto the modulator amplifier, and f is the scaling of the feedback gain, i. e. it measures therelation between the intensity incident on the photodiode and the voltage delivered by thedifferential amplifier. In general, B is used as a control parameter.In the case of a single laser, the phase of the electric field is a constant variable in timeand has no role in the system dynamics [23]. Thus, we prefer to work with the amplitude of3 | E | PB H
FIG. 1. High-gain bifurcation diagram. The stationary amplitude of the laser field | E | is shown asa funtion of the bias voltage B . The solid and dashed lines mark the stable and unstable branches,respectively, while the arrows indicate the hysteresis loop. H denotes the Hopf bifurcation pointand PB the subcritical pitchfork bifurcation. The constant value of B = 0 .
37 that will be usedin the following sections has been indicated by the arrow and the red line. Other parameters are γ = 0 . P = 1 . β = 0 . a = 5 .
8, and f = − .
6. Dotted line denotes a very smallregime of low-amplitude oscillations. the electric field without loss of generality. In this framework, the system of Eqs. (1) admitsthe zero-intensity solution ( | E | = 0 , G = P, V = B ) and the non-zero intensity solution(s)which are given in the parametric form P | E | = 1 + a sin (cid:0) B + f | E | (cid:1) (2a) G = P | E | V = B + f | E | . (2b)Figure 1 illustrates the stability of these fixed points by studying the bifurcation diagram4n the case of high gain f = − . B as the control parameter [10].For B < . B = 0 . B = 0 . . < B < . B increases beyond the value 0 . B = 0 .
37 in order to achieve a controllable bistablesystem that can be prepared either in the passive state 0 < | E | < . . < | E | < .
9. Moreover, the chosen value B = 0 .
37 allows us to avoid transitions fromthe Hopf point at B = 0 . III. TWO COUPLED BISTABLE LASERS
Having defined the single bistable laser system, we proceed by considering two parallelwaveguides of CO lasers each one with a proper optoelectronic feedback (see Fig. 2). Themutual interaction lies on the overlap integrals of both lasers fields inside the crystal with aproper refractive index profile [9]. The evolution equations for this coupled system have theform dEdt = E G − − a sin ( V )) + ηE H (3a) dE H dt = E H G H − − a sin ( V H )) + ηE , (3b)where the subscript H denotes the second laser. The equations for the population inversion( G and G H ) and the feedback voltage of the modulator ( V and V H ) have the same form as inEqs. 1, therefore we omit them. The parameter η is the coupling strength between the twolasers and in general is a complex parameter ( η = η (cid:60) + iη (cid:61) ). The real part η (cid:60) takes usually5 eam splitter crystal amplifier p h o t o d i o d e p h o t o d i o d e amplifierLaser 1Laser 2 a b LaserLaserLaser Laser LaserLaserHub.... N j N N N N N FIG. 2. a) Schematic diagram of the optoelectronic feedback of two coupled lasers. The opticalpower emitted by the two lasers is coupled through the overlap of the electrical fields in a nonlinearcrystal. After a beam splitter, it is detected by a photodiode with a fixed bandwidth. The electricaloutput is fed back to each laser through an amplifier. b) Topology of a star network where eachlaser of the periphery interacts with the rest through a central laser, the hub, with tha samecoupling strength. negative values and vanishes only when D (cid:39) w , where D is the distance between the twobeams and w is the waist of the beam Gaussian portrait. However, it is possible to havepositive coupling values, which we consider here, by pumping in the middle between the twobeams [25]. The imaginary part η (cid:61) , is related to the refractive index and can be zero for alaser beam of weak intensity, which is the case here. If we use polar coordinates E = | E | e iφ ,a third equation for the phase difference of the two lasers is added to Eqs. 3. However, wecan neglect the dynamics of the third variable since we are working in the phase lockingregime, i. e. the phase difference is constant and equal to zero (See Fig. S1 in SupplementalMaterial (SM) [26]). The dynamics of the system can, therefore, be described solely by theamplitude of the electric field.The zero intensity steady state of the coupled system is equal to | E | = | E H | = 0 , G = G H = P, V = V H = B , while the non-zero intensity steady states of Eqs. 3, are given in the6 IG. 3. The stationary amplitude of the laser field versus the coupling strength. a) The amplitudeof the first laser in the case where the system is prepared with the first laser located in the passivestate and the second in the active state (passive-active). b) The stationary amplitude of thelaser field versus the coupling strength where both the two lasers are located in the active state(active-active). c) The stability for all three preparation states of the system (passive-active, active-active and passive-passive). d) The stability region for the amplitude of both lasers in all threecases is shown in the ( | E | , | E H | ) plane. Solid and dashed lines correspond to stable and unstablesteady states, respectively. SN stands for the saddle-node bifurcation, while PB for the subcriticalpitchfork. B = 0 .
37 and all other parameters as in Fig. 1. parametric form P | E | = 1 + a sin (cid:0) B + f | E | (cid:1) + 2 η | E H || E | (4a) P | E H | = 1 + a sin (cid:0) B + f | E H | (cid:1) + 2 η | E || E H | . (4b)Figure 3 shows the stability of the system steady states as a function of the couplingstrength. Figure 3 (a) shows the stationary amplitude of the electric fields in the case where7he system is prepared with the first laser in the passive and the second one in the activestate. Similarly, Fig. 3 (b) shows the stability of the system when both lasers are preparedin the active state. In Fig. 3 (c) the two previous cases (passive-active, active-active) areplotted together with the passive-passive state which corresponds to the black line. Figure 3(d) shows the stability region for the amplitude of both lasers in all three cases is shown inthe ( | E | , | E H | ) plane, with the passive-passive state represented by a full black circle. Thethick and dashed lines correspond to the stable and unstable solution branches, respectively.From Fig. 3 (c) we can see that the passive-passive state (black line) undergoes a subcriticalpitchfork bifurcation (PB) at η = 0 . η = 0 . η range and coexists withan unstable branch that emerges at the PB point and runs through the negative axis (notshown here because η has physical meaning only for positive values).As a result of the above stability analysis, when the system starts at the passive-activestate, both lasers jump to the active state (green branch) through a SN bifurcation at ratherlow coupling strengths η > . η > . IV. STAR NETWORK OF COUPLED BISTABLE LASERS
Having analyzed the dynamics of two coupled bistable lasers, we now focus on a starnetwork configuration and how it contributes to the formation mechanism of stationaryactive patterns. In such a system each element of the periphery interacts with the restthrough a central element, the hub (see Fig, 2 (b)), thus Eqs. (3) can be reformulated as8ollows: dE j dt = E j G j − − a sin ( V j )) + ηE H (5a) dE H dt = E H G H − − a sin ( V H )) + η N (cid:88) j =1 E j , (5b)where j = 1 , . . . N counts for the number N of the peripheral elements and the subscript H denotes the hub. In polar coordinates Eqs. 5 become: d | E j | dt = 12 | E j | (cid:2) G j − − a sin (cid:0) V j )] + η | E H | cos( θ j ) (6a) d | E H | dt = 12 | E H | (cid:2) G H − − a sin (cid:0) V H )] + η N (cid:88) j =1 | E j | cos( θ j ) (6b) dθ j dt = − η (cid:34) | E H || E j | sin( θ j ) + N (cid:88) k =1 | E k || E H | sin( θ k ) (cid:35) , (6c)where θ j = φ H − φ j are the phase differences between the electric fields of each node of theperiphery and that of the hub. The equations for the variables G j , V j , G H , V H have thesame form with Eqs. (1) and, again, we omit them. Numerical integration of Eqs. (6) showsthat in the N − η parameter space the phase differences θ j remain constant and equal tozero for η > .
002 (see Fig. S2 in the Supplemental Material (SM) [26]). Therefore, Eq. (6c)can be neglected, the cosine terms are equal to 1, and the index j can be dropped, reducingthe star network to a system of two coupled lasers with asymmetric coupling. d | E | dt = 12 | E | (cid:2) D − − a sin (cid:0) V )] + η | E H | (7a) d | E H | dt = 12 | E H | (cid:2) D H − − a sin (cid:0) V H )] + ηN | E | . (7b)Previous studies with electrochemical systems [19, 20, 27] have implemented similar methodsfor reducing star and tree networks to chains of asymmetrically coupled nodes. In those9 .002 0.01 0.02 0.03 0.04 0.05135710 i ii iii ic ivi ii iiiic i ii iii iv FIG. 4. Phase diagram in the ( η, N ) parametric space. Four dynamical regions are separatedby curves that correspond to the continuation of the bifurcation points shown in Fig. 3 (orangeand red curves correspond to saddle-node bifurcation lines, while the black curve corresponds to apitchfork bifurcation line). In region I the coupling is weak enough and all three initial conditions(IC) shown in the inset are stable and consist steady states of the system. In region II the activeperiphery drifts the hub to the active state. In region III the active periphery drifts the hub to theactive state but also the active hub drifts the periphery in the active state. In region IV the wholenetwork goes to the active state. In the inset the inner circle represents the hub and the outercircle represents the periphery, while the active state is denoted with blue color and the passivestate with yellow. Other parameters as in Fig. 3. theoretical and experimental studies, it was demonstrated that such a reduced system couldproduce all the rich dynamics of the original network despite its simpler form.Again, the zero intensity solution corresponds to | E | = | E H | = 0 , G = G H = P, V =10 H = B and the non-zero intensity solutions are given in the parametric form P | E | − − a sin (cid:0) B + f | E | (cid:1) = − η | E H || E | (8a) P | E H | − − a sin (cid:0) B + f ε H (cid:1) = − ηN | E || E H | . (8b)In the previous section, the system of two symmetrically coupled lasers has shown that,depending on the preparation of the coupled system, the lasers transition to the active stateoccurs either through a saddle-node or a subcritical pitchfork bifurcation. In order to locatethese transitions in the star-network, we perform a continuation of the bifurcations in the( η, N ) parameter space as shown in Fig. 4. The line (orange color) separating regions I andII corresponds to the continuation of the SN bifurcation in the case where the hub sartsin the passive and the periphery in the active state. Our reduced system with the twoasymmetrically coupled lasers is directed, therefore the passive-active state stability shouldbe considered for the opposite case as well, i. e. for the hub in the active state and theperiphery in the passive. This latter bifurcation line (red color) starts at the same couplingstrength value as the previous bifurcation line, but has a different behavior and separates theregions II and III. Finally, the line (black color) separating regions III and IV correspondsto the continuation of the PB bifurcation that marks the transition from the passive-passiveto the active-active state.These bifurcation lines separate the ( η, N ) parameter space in four distinct regions wherethe system reaches different steady states. In the region I, any initial condition (peripheryactive-hub passive, periphery passive-hub active, or periphery passive-hub passive) remainsas it is, namely the system is pinned to its initial preparation. In region II, the activeperiphery drifts the passive hub into the active state. The same occurs in region III where,additionally, the active hub drifts the passive periphery into the active state. In this regionthe activation propagates faster from the active periphery towards the passive hub, thanfrom an active hub towards the periphery. Finally, in region IV the coupling strength isstrong enough even for the periphery passive-hub passive initial condition to jump to theactive-active state. An example of the described dynamical behaviors is shown in he inset ofFig. 4, where the evolution of three initial conditions (IC) in the ( η, N ) parameter space is11llustrated. The outer circle represents the periphery of the system, the inner circle representsthe hub, while light and dark colors correspond to the passive and active states, respectively.In close analogy to previous findings in electrochemical bistable networks [19, 20] Fig. 4shows that the coupling strength required for a transition to occur in the system’s dynamicsdepends on the number of the peripheral nodes. The line (orange color) separating regionsI and II drops with η because as the coupling strength increases, a smaller N size is neededfor the periphery to activate the hub (and vice-versa). This results in a shift to lower η values, of the position of the saddle-node bifurcation when N increases (see Fig. S3 (b) ofthe Supplemental Material). On the other hand, the number of periphery nodes is almost(red line has a tiny slope) independent of the coupling strength required for the hub toactivate the periphery (red line and Fig. S3 (c) of the Supplemental Material). Finally, theline separating regions III and IV (black color), which marks the activation of both passivehub and passive periphery, exists for higher values of the coupling strength and, similarlyto the orange line, drops with η (see Fig. S3 (d) of the Supplemental Material). V. CONCLUSIONS
We have shown that star networks of coupled bistable class B lasers support activationspreading from the hub towards the peripheral elements and vice versa. Interestingly, sta-tionary patterns of activation localized on the hub or the peripheral nodes are also supported,determined by the number of coupled lasers to the central unit, by the coupling strength,and the initial conditions. Similar findings were previously reported for electrochemical sys-tems. However, the system considered in the current work has been implemented for coupledCO lasers with optoelectronic feedback keeping the bias voltage applied to the modulatorconstant and by considering the coupling strength as a control parameter. After carefulnumerical calculations, the phases of the central laser and any peripheral unit lock after avery small time interval allowing us to investigate only the steady state of the system.In a system size-coupling strength diagram we demonstrate four distinct regions indicatingdifferent dynamical behavior. At weak coupling strengths and small network sizes the initialpreparation of the system is pinned and an activation remains stationary and localizedeither on the peripheral elements or on the hub. At weak coupling strengths but largernetwork sizes an activation can spread only from the periphery towards the hub but not in12he the opposite direction. Namely an activated periphery turns on the center element butan activated hub cannot drift the periphery to the active state. This occurs for moderatevalues of the coupling strength. In this third region activation spreads in both directions(with different velocities) and an activated periphery turns on the hub as well as an activatedhub can drift the periphery into the active state. Finally, an activation of the whole systemfrom the passive into the operative region (active state) is shown for strong couplings.Despite the obviously different nature of the considered system with the previously studiedelectrochemical networks, our findings have essential similarities indicating that the networkconnectivity affects the hosted bistable dynamics in an akin fashion. The ability to controlthe spreading or the pinning of an activation, thus the dynamics of the system from thepassive into the active states and vice versa, may have multiple technological applicationsespecially in neuromorphic photonics [28, 29], where such tree-like networks can serve forsimple hierarchical connectivity structures. For future studies, it would be worthwhile toexplore if those stationary states can live in the presence of small phase perturbations, dueto spontaneous emission or through the detuning of each individual laser cavity length.Moreover, it would be interesting to consider the bias regime where the systems exhibitsoscillations, and instead of the CO laser to study a semiconductor laser diode with asaturable absorber. VI. ACKNOWLEDGMENTS
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